Exercises

1. Consider an electromagnetic wave propagating through a nonuniform dielectric medium whose dielectric constant $\epsilon$ is a function of ${\bf r}$ . Demonstrate that the associated wave equations take the form

   $$\nabla^{\,2}{\bf E} - \frac{\epsilon}{c^{\,2}}\,\frac{\partial^{\,2}{\bf E}}{\partial t^{\,2}} =- \nabla\left(\frac{\nabla\epsilon\cdot{\bf E}}{\epsilon}\right),$$

   $$\nabla^{\,2}{\bf B} - \frac{\epsilon}{c^{\,2}}\,\frac{\partial^{\,2}{\bf B}}{\partial t^{\,2}} = - \frac{\nabla\epsilon\times (\nabla\times {\bf B})}{\epsilon}.$$

   
   

2. Suppose that a light-ray is incident on the front (air/glass) interface of a uniform pane of glass of refractive index $n$ at the Brewster angle. Demonstrate that the refracted ray is also incident on the rear (glass/air) interface of the pane at the Brewster angle.

3. Consider an electromagnetic wave obliquely incident on a plane boundary between two transparent magnetic media of permeabilities $\mu_1$ and $\mu_2$ . Find the coefficients of reflection and transmission as functions of the angle of incidence for the wave polarizations in which all electric fields are parallel to the boundary and all magnetic fields are parallel to the boundary. Is there a Brewster angle? If so, what is it? Is it possible to obtain total reflection? If so, what is the critical angle of incidence required to obtain total reflection?

4. A medium is such that the product of the phase and group velocities of electromagnetic waves is equal to $c^{\,2}$ at all wave frequencies. Demonstrate that the dispersion relation for electromagnetic waves takes the form
   $$\omega^{\,2} = k^{\,2}\,c^{\,2}+\omega_0^{\,2},$$
   where $\omega_0$ is a constant.

5. Demonstrate that if the equivalent height of reflection in the ionosphere varies with the angular frequency of the wave as
   $$h(\omega)= h_0 + \delta\left(\frac{\omega}{\omega_0}\right)^p,$$
   where $h_0$ , $\delta$ , and $\omega_0$ are positive constants, then $\omega_p(z)=0$ for $z<h_0$ , and
   $$\omega_p(z) =\left[\frac{\pi\,{\mit\Gamma}(1+p)}{{\mit\Gamma}(1/2... ...2+p/2)}\right]^{1/p} \frac{\omega_0}{2}\left(\frac{z-h_0}{\delta}\right)^{1/p}$$
   for $z\geq h_0$ . Here, ${\mit\Gamma}(z)$ is a Gamma function.

6. Suppose that the refractive index, $n(z)$ , of the ionosphere is given by $n^2=1-\alpha\,(z-h_0)$ for $z\geq h_0$ , and $n^2=1$ for $z<h_0$ , where $\alpha$ and $h_0$ are positive constants, and the Earth's magnetic field and curvature are both neglected. Here, $z$ measures altitude above the Earth's surface.
   1. A point transmitter sends up a wave packet at an angle $\theta$ to the vertical. Show that the packet returns to Earth a distance
      $$x_0 = 2\,h_0\,\tan\theta + \frac{2}{\alpha}\,\sin2\theta$$
      from the transmitter. Demonstrate that if $\alpha\,h_0<1/4$ then for some values of $x_0$ the previous equation is satisfied by three different values of $\theta$ . In other words, wave packets can travel from the transmitter to the receiver via one of three different paths. Show that the critical case $\alpha\,h_0=1/4$ corresponds to $\theta=\pi/3$ and $x_0=6\sqrt{3}\,h_0$ .
   2. A point radio transmitter emits a pulse of radio waves uniformly in all directions. Show that the pulse first returns to the Earth a distance $4\,h_0\,(2/\alpha\,h_0-1)^{1/2}$ from the transmitter, provided that $\alpha\,h_0<2$ .